DR R. A. SAMPSON ON A 



radius greater than 1 <l.-^rt- \v-iv desired, it could be made even smaller, but it would 

 aeem t involve tin- sacrifice of some other conveniences. 



Tli.- v.-dii.-s uf tin- quantities <J,G, &c., for the combined system are built up step by 

 step by proceeding from surface to surface or from lens to lens by the sequence 

 qua ti. His (17), p. 160, of the Memoir referred to above. For making these steps 

 it is not convenient to lay down any one procedure as being the best for all cases, 

 but two methods may be mentioned, one or other of which is frequently suitable. 

 t we can proceed from conjugate focus to conjugate focus, the first focus being 

 the principal focus of the first or great mirror, and each successive conjugate focus 

 being the principal focus of the whole combination which precedes it. That is to say, 



at each stage we have 



g = 0, hk= -1, h' = 0, 



so that the equations we require to consider are 



. (4) 



In these fir', ... refers to the new or added element, g, ... to the combination from 

 the beginning up to this element, and G, ... to the resulting combination including 

 this element. We thus notice that S,g contributes to S,G simply by multiplying by 

 flr', which is the magnification of the new element between its conjugate foci under 

 consideration. We notice, too, that so long as we confine ourselves to ,G, the only 

 coefficients which it is necessary to find for each added element are SJi', calculated 

 between the same conjugate foci. If the aberrations of the second element are given, 

 referred to some other origins, they must be transferred to the conjugate foci in 

 question by means of the equations for change of origin (22), p. 164. A case will 

 present itself that requires a modification of this process, namely, when one of the 

 conjugate foci belonging to an element introduced by one of the steps described is at 

 a great distance ; to meet this case we may take this element together with the next 

 following m,e and combine them into one before adding them to the combination, or 

 we may take a second completely different method as follows : 



Let O w O. be the initial and final origins ; O , O . the origins to which the known 

 aberrations of a part of the system are referred. Calling {g', h' ; k', I'} the subsequent 

 normal system O.- to O., transfer the aberrations to origins O a ...O B by use of the first 

 part of ,.,,,,ati,,,,H (17), p. 160, viz., S,G = g'S l g + h'S l k ..... Then caUing {g, h ; k, 1} 



pi-.-.-,- ling MMr.ual scheme O to O a , transfer the so-found coefficients from O.....O,, 

 ( >. by using the forms of the second part of the same equations. An example 

 ' this method will be found on p. 55. 



We now study the formulae for thin lenses. It will be pointed out later how to 

 make use of these when the lenses are thick. 



